Nuprl Lemma : bounded-expectation
∀p:FinProbSpace. ∀f:ℕ ─→ ℕ. ∀X:n:ℕ ─→ RandomVariable(p;f[n]). ∀B:ℚ.
(nullset(p;(X[n]─→∞ as n─→∞))) supposing
((∀n:ℕ. (0 ≤ X[n] ∧ E(f[n];X[n]) < B)) and
0 < B and
(∀n:ℕ. ∀i:ℕn. X[i] ≤ X[n]) and
(∀n:ℕ. ∀i:ℕn. f[i] < f[n]))
Proof
Definitions occuring in Statement :
rv-unbounded: (X[n]─→∞ as n─→∞),
nullset: nullset(p;S),
rv-le: X ≤ Y,
expectation: E(n;F),
rv-const: a,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ─→ B[x],
natural_number: $n,
rationals: ℚ
Lemmas :
member-less_than,
int_seg_subtype-nat,
false_wf,
nat_wf,
int_seg_wf,
rv-le_witness,
subtype_rel-random-variable,
le_weakening2,
Error :qless_witness,
int-subtype-rationals,
rv-const_wf,
expectation_wf,
all_wf,
rv-le_wf,
Error :qless_wf,
less_than_wf,
rationals_wf,
random-variable_wf,
finite-prob-space_wf,
Error :qless_transitivity_2_qorder,
Error :qle_weakening_eq_qorder,
Error :qless_irreflexivity,
equal-wf-T-base,
Error :qmul_preserves_qless,
qdiv_wf,
Error :qinv-positive,
qmul_wf,
squash_wf,
true_wf,
Error :qmul_comm_qrng,
qinv_wf,
assert-qeq,
assert_wf,
qeq_wf2,
not_wf,
iff_weakening_equal,
Error :qmul_zero_qrng,
Error :qmul_assoc_qrng,
Error :qmul_one_qrng,
Markov-inequality,
rv-qle_wf,
equal_wf,
assert_functionality_wrt_uiff,
qmul_com,
Error :qdiv-qdiv,
Error :qmul-qdiv-cancel2,
qmul_ident,
Error :qless_transitivity_1_qorder,
exists_wf,
p-open_wf,
p-measure-le_wf,
p-outcome_wf,
Error :qle_wf,
subtype_rel_dep_function,
length_wf,
p-open-member_wf,
decidable__exists_int_seg,
subtype_rel-int_seg,
decidable__cand,
decidable__lt,
Error :decidable__qle,
decidable_wf,
lelt_wf,
equal-wf-base,
iff_wf,
le_wf,
subtype_rel_self,
decidable__equal_int,
less_than_transitivity2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-swap,
add-commutes,
le-add-cancel,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
decidable__le,
subtract_wf,
not-ge-2,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-zero,
not-le-2,
subtract-is-less,
subtype_rel_set,
Error :qle_reflexivity,
expectation-monotone-in-first,
expectation-monotone,
subtype_base_sq,
int_subtype_base,
q_le_wf,
bool_wf,
bnot_wf,
Error :qle-int,
uiff_transitivity2,
eqtt_to_assert,
Error :assert-q_le-eq,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
Error :qle_transitivity_qorder,
add-mul-special,
zero-mul,
sq_stable_from_decidable,
Error :decidable__qless,
rv-unbounded_wf,
set_wf,
le_weakening
\mforall{}p:FinProbSpace. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n]). \mforall{}B:\mBbbQ{}.
(nullset(p;(X[n]{}\mrightarrow{}\minfty{} as n{}\mrightarrow{}\minfty{}))) supposing
((\mforall{}n:\mBbbN{}. (0 \mleq{} X[n] \mwedge{} E(f[n];X[n]) < B)) and
0 < B and
(\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. X[i] \mleq{} X[n]) and
(\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. f[i] < f[n]))
Date html generated:
2015_07_17-AM-08_01_56
Last ObjectModification:
2015_02_03-PM-09_47_36
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